Let A be an mxn matrix with entries from a field.
Our objective is to prove that the row-space and column-space of A have that same dimension i.e. the same rank.
a) https://math.stackexchange.com/a/2254859/231663
b) As a follow on from part (a) Suppose that A is in reduced echelon form. Use part (a) to show that its row-space and column-space have the same dimension.
c) Show that if A′ is obtained from A by an elementary row operation, then the row-space of A′ is equal to the row-space of A.
d) Give an example showing that if A′ is obtained from A by an elementary row operation, then the column- space of A′ is not necessarily equal to the column-space of A.
e) Suppose that A′ is obtained from A by an elementary row operation. Show that a collection of columns of A is linearly independent if and only if the corresponding collection of columns in A′ is linearly independent. More precisely show that if c1, . . . , cn are the columns of A, c′1, . . . , c′n are the columns of A′ and J ⊆ {1,...,n}, then {ci : i ∈ J} is linearly independent if and only if {c′i : i ∈ J} is linearly independent.
f) Conclude that the row-space and column-space of A have the same dimension.
